Class groups and Brauer groups

LetF be a global field,n a positive integer not divisible by the characteristic ofF. Then there exists a finite extensionE ofF whose class group has a cyclic direct summand of ordern. This theorem, in a slightly stronger form, is applied to determine completely, on the basis of the work of Fein and Schacher, the structure of the Brauer group Br(F()) of the rational function fieldF(t). As a consequence of this, an additional theorem of the above authors, together with a note at the end of the paper, imply that Br(F(t)) ≊ Br(F(t ₁, ···,t _n)), wheret ₁, ···,t _n are algebraically independent overF.     

Automorphism groups of polycyclic-by-finite groups and arithmetic groups

We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ,1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory.     

Automorphism groups of nilpotent groups

We construct a full class of nilpotent groups of class 2 of an arbitrary infinite cardinality . Their centers, commutator subgroups and factors modulo the center will be the same and a homogeneous direct sum of a group of rank 1 or 2. Their automorphism groups will coincide and the factor group modulo the stabilizer could be an arbitrary group of size $\leqq$ .     

Brauer groups,embedding problems,and nilpotent groups as Galois groups

Let ℚ _ab denote the maximal abelian extension of the rationals ℚ, and let ℚ_abnil denote the maximal nilpotent extension of ℚ _ab . We prove that for every primep, the free pro-p group on countably many generators is realizable as the Galois group of a regular extension of ℚ_abnil(t). We also prove that ℚ_abnil is not PAC (pseudo-algebraically closed). This paper was inspired by the author's participation in a special year on the arithmetic of fields at the Institute for Advanced Studies at the Hebrew University of Jerusalem. I would like to express my appreciation to the Institute for its hospitality, and to the organizers, especially Moshe Jarden. Partially supported by the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund-Japan Technion Society Research Fund.     

Quotient groups of locally graded groups and groups of certain Kurosh-Chernikov classes

We establish the validity of the inclusion G/N∈ X for groups G ∈ X under certain restrictions on N ⊴ G, where X is one of the following classes, the class of locally graded groups, the class of RI-groups, or the class for a fixed group variety . Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1545–1553, November, 1998.     

Endomorphisms of automorphism groups of free groups

It is proved that any non-trivial endomorphism of an automorphism group AutF_n of a free group F_n, for n 3, either is an automorphism or factorization over a proper automorphism subgroup. An endomorphism of AutF₂ is an automorphism, or else a homomorphism onto one of the groups S₃, D₈, Z₂ × Z₂, Z₂, or (Z₂ × Z₂). A non-trivial homomorphism of AutF_n into AutF_m, for n 3, m 2, and n > m, is a homomorphism onto Z₂ with kernel SAutF_n. As a consequence, we obtain that AutF_n is co-Hopfian.Supported by RFBR grant No. 02-01-00293 and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 211–237, March–April, 2005.     

Automorphism groups of nilpotent groups and spaces

A pair of finitely generated, torsion-free nilpotent groups G₁,G₂ is constructed with the properties that G₁ and G₂ are p-isomorphic for all primes p, yet Aut(G₁) and Aut(G₂) are not isomorphic. The example constructed is compared to an analogous example in the homotopy category of simply connected, finite CW-complexes.     

On automorphism groups of connected Lie groups

We prove that ifG is a connected Lie group with no compact central subgroup of positive dimension then the automorphism group ofG is an almost algebraic subgroup of , where is the Lie algebra ofG. We also give another proof of a theorem of D. Wigner, on the connected component of the identity in the automorphism group of a general connected Lie group being almost algebraic, and strengthen a result of M.Goto on the subgroup consisting of all automorphisms fixing a given central element.     

Bounding Ext for modules for algebraic groups, finite groups and quantum groups

Given a finite root system Φ, we show that there is an integer c=c(Φ) such that , for any reductive algebraic group G with root system Φ and any irreducible rational G-modules L, L^′. There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for Extn, for any integer n?0, using a constant depending only on n and the root system. When L is the trivial module, the same result is proved in the algebraic group case, thus giving similar bounded properties, independent of characteristic, for algebraic and generic cohomology. (A similar result holds for any choice of L=L(λ), even allowing λ to vary, provided the p-adic expansion of lambda is limited to a fixed number of terms.) In particular, because of the interpretation of generic cohomology as a limit for underlying families of finite groups, the same boundedness properties hold asymptotically for finite groups of Lie type. The results both use, and have consequences for, Kazhdan–Lusztig polynomials. Appendix A proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.     

Definable groups and compact p-adic Lie groups

We formulate p-adic analogues of the o-minimal group conjecturesfrom the works of Hrushovski, Peterzil and Pillay [J. Amer.Math. Soc., to appear] and Pillay [J. Math. Log. 4 (2004) 147–162];that is, we formulate versions that are appropriate for groupsG definable in (saturated) P-minimal fields. We then restrictour attention to saturated models K of Th(_p) and Th(_{p, an}),record some elementary observations when G is defined over thestandard model _p, and then make a detailed analysis of the casewhere G = E(K) for E an elliptic curve over K. Essentially,our P-minimal conjectures hold in these contexts and, moreover,our case study of elliptic curves yields counterexamples toa more naive direct translation of the o-minimal conjectures.     

Infinite locally dihedral groups as automorphism groups

If \(A\) is a nontrivial torsion-free, locally cyclic group with no nontrivial divisible quotients, and \(G\) is the split extension of \(A\) by a group of order 2 acting on \(A\) by means of the inverting map, then \(G\simeq {{{\mathrm{Aut}}}G} \). We prove that in no other case the full automorphism group of a group is infinite and locally dihedral.     

On braid groups and right-angled Artin groups

In this article we prove a special case of a conjecture of A. Abrams and R. Ghrist about fundamental groups of certain aspherical spaces. Specifically, we show that the \(n\) -point braid group of a linear tree is a right-angled Artin group for each \(n\) .     

On outer automorphism groups of coxeter groups

Summary It is shown that the outer automorphism group of a Coxeter groupW of finite rank is finite if the Coxeter graph contains no infinite bonds. A key step in the proof is to show that if the group is irreducible andΠ ₁ andΠ ₂ any two bases of the root system ofW, thenΠ ₂ = ±ωΠ ₁ for some ω εW. The proof of this latter fact employs some properties of the dominance order on the root system introduced by Brink and Howlett. This article was processed by the author using the Springer-Verlag TEX PJour1g macro package 1991.     

Ordering groups and validity in lattice-ordered groups

An inductive characterization is given of the subsets of a group that extend to the positive cone of a right order on the group. This characterization is used to relate validity of equations in lattice-ordered groups (?-groups) to subsets of free groups that extend to the positive cone of a right order. As a consequence, new proofs are obtained of the decidability of the word problem for free ?-groups and generation of the variety of ?-groups by the ?-group of automorphisms of the real line. An inductive characterization is also given of the subsets of a group that extend to the positive cone of an order on the group. In this case, the characterization is used to relate validity of equations in varieties of representable ?-groups to subsets of relatively free groups that extend to the positive cone of an order.     

On automorphism groups of some finite groups

We show that if n > 1 is odd and has no divisor p4 for any prime p, then there is no finite group G such that│Aut(G)│ = n.     

On coleman outer automorphism groups of finite groups

Let G be a finite group and Out _Col(G) the Coleman outer automorphism group of G(for the definition, see below). The question whether Out_Col(G) is a p′-group naturally arises from the study of the normalizer problem for integral group rings, where p is a prime. In this article, some sufficient conditions for Out_Col(G) to be a p′-group are obtained. Our results generalize some well-known theorems.     

Automorphism groups of free metabelian nilpotent groups

Published in Algebra i Logika, Vol. 29, No. 6, pp. 746–751, November–December, 1990.     

Chow-Witt groups and Grothendieck-Witt groups of regular schemes

Let A be a noetherian commutative Z[1/2]-algebra of Krull dimension d and let P be a projective A-module of rank d. We use derived Grothendieck-Witt groups and Euler classes to detect some obstructions for P to split off a free factor of rank one. If d?3, we show that the vanishing of its Euler class in the corresponding Grothendieck-Witt group is a necessary and sufficient condition for P to have a free factor of rank one. If d is odd, we also get some results in that direction. If A is regular, we show that the Chow-Witt groups defined by Morel and Barge appear naturally as some homology groups of a Gersten-type complex in Grothendieck-Witt theory. From this, we deduce that if d=3 then the vanishing of the Euler class of P in the corresponding Chow-Witt group is a necessary and sufficient condition for P to have a free factor of rank one.     

On alternating and symmetric groups as Galois groups

Fix an integern≧3. We show that the alternating groupA _n appears as Galois group over any Hilbertian field of characteristic different from 2. In characteristic 2, we prove the same whenn is odd. We show that any quadratic extension of Hilbertian fields of characteristic different from 2 can be embedded in anS _n-extension (i.e. a Galois extension with the symmetric groupS _n as Galois group). Forn≠6, it will follow thatA _n has the so-called GAR-property over any field of characteristic different from 2. Finally, we show that any polynomialf=X ⁿ+…+a₁X+a₀ with coefficients in a Hilbertian fieldK whose characteristic doesn’t dividen(n-1) can be changed into anS _n-polynomialf ^* (i.e the Galois group off ^* overK Gal(f ^*, K), isS _n) by a suitable replacement of the last two coefficienta ₀ anda ₁. These results are all shown using the Newton polygon. The author acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2000-00114, GTEM.